Battery state and parameter estimation using a mixed sigma-point kalman filtering and recursive least squares technique

ABSTRACT

A battery management system for an electrified powertrain of a hybrid vehicle includes one or more sensors configured to measure voltage, current, and temperature for a battery system of the hybrid vehicle and a controller. The controller is configured to obtain an equivalent circuit model for the battery system, determine a set of states for the battery system to be estimated, determine a set of parameters for the battery system to be estimated, receive, from the sensor(s), the measured voltage, current, and temperature for the battery system, using the equivalent circuit model and the measured voltage, current, and temperature of the battery system, estimate the sets of states and parameters for the battery system using a mixed sigma-point Kalman filtering (SPKF) and recursive least squares (RLS) technique, and using the sets of estimated states/parameters for the battery system, control an electric motor of the electrified vehicle.

FIELD

The present application generally relates to a battery management systemfor an electrified powertrain of a hybrid vehicle.

BACKGROUND

Some hybrid vehicles include an electrified powertrain comprising anelectric motor that is powered by a battery system. Such hybrid vehiclesinclude battery electric vehicles (BEV), plug-in hybrid electricvehicles (PHEV), and the like. These hybrid vehicles include batterymanagement systems (BMS) configured to monitor the battery system. Thismonitoring typically includes estimating states of the battery system(state of charge (SOC), voltages, etc.) and parameters of the batterysystem (resistances, time constants, etc.).

These states and parameters are also utilized to determine otherstates/parameters, such as the state of power (SOP) of the batterysystem and/or the state of health (SOH) of the battery system. Linearmodels are often used for this state/parameter estimation, but thesemodels are typically inaccurate. Non-linear models could be usedinstead, but these models are typically difficult to tune. Accordingly,while such BMS work for their intended purpose, there remains a need forimprovement in the relevant art.

SUMMARY

According to an aspect of the invention, a battery management system foran electrified powertrain of a hybrid vehicle is presented. In oneexemplary implementation, the system includes one or more sensorsconfigured to measure voltage, current, and temperature for a batterysystem of the hybrid vehicle and a controller configured to: obtain anequivalent circuit model for the battery system; determine a set ofstates for the battery system to be estimated; determine a set ofparameters for the battery system to be estimated; receive, from the oneor more sensors, the measured voltage, current, and temperature for thebattery system; and using the equivalent circuit model and the measuredvoltage, current, and temperature of the battery system, estimate thesets of states and parameters for the battery system using a mixedsigma-point Kalman filtering (SPKF) and recursive least squares (RLS)technique; and using the sets of estimated states/parameters for thebattery system, control an electric motor of the electrified vehicle.

In some implementations, the controller is configured to estimate: theset of states for the battery system using an SPKF technique; and theset of parameters for the battery system using an RLS technique. In someimplementations, the controller is further configured to estimate: theset of parameters for the battery system estimated using the RLStechnique are an input for the SPKF technique; and the set of states forthe battery system estimated using the SPKF technique are an input forthe RLS technique. In some implementations, the set of states includesat least one of a state of charge (SOC) of the battery system and avoltage of the battery system.

In some implementations, the SPKF technique and the RLS technique areboth configured to estimate a particular state or parameter for thebattery system, and wherein the controller is configured to estimate theparticular state or parameter of the battery system based on at leastone of these estimates. In some implementations, the controller isfurther configured to calculate a covariance indicative of a confidencein each estimate of the particular state or parameter for the batterysystem, wherein the controller is configured to utilize the covariancein estimating the particular state or parameter of the battery system.In some implementations, the controller is configured to estimate theparticular state or parameter for the battery system based on an averageof the estimates from the SPKF and RLS techniques.

In some implementations, the controller is configured to estimate theset of parameters for the battery system using the RLS technique basedfurther on a tuned forgetting factor. In some implementations, the setof parameters includes at least one of a resistance and acharge/discharge time constant for the battery system. In someimplementations, the equivalent circuit model is an asymmetricequivalent circuit model having charge/discharge asymmetry.

In some implementations, the controller is further configured to: obtainan SPKF rate at which to perform the SPKF technique; obtain an RLS rateat which to perform the RLS technique; and estimate the sets of statesand parameters for the battery system using the SPKF and RLS techniquesaccording to their respective rates. In some implementations, the SPKFrate is greater than the RLS rate.

In some implementations, the controller is further configured toestimate at least one of a state of power (SOP) and a state of health(SOH) for the battery system based on the sets of estimated states andparameters. In some implementations, the controller is configured tocontrol the electric motor based further on the at least one of the SOPand the SOH for the battery system. In some implementations, the hybridvehicle comprises an engine; and the controller is configured tocoordinate control of the engine and the electric motor based on theestimated states/parameters to maximize usage of the electric motor.

Further areas of applicability of the teachings of the presentdisclosure will become apparent from the detailed description, claimsand the drawings provided hereinafter, wherein like reference numeralsrefer to like features throughout the several views of the drawings. Itshould be understood that the detailed description, including disclosedembodiments and drawings referenced therein, are merely exemplary innature intended for purposes of illustration only and are not intendedto limit the scope of the present disclosure, its application or uses.Thus, variations that do not depart from the gist of the presentdisclosure are intended to be within the scope of the presentdisclosure.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a functional block diagram of an example hybrid vehicleaccording to the principles of the present disclosure;

FIG. 2 is a functional block diagram of an example configuration for acontroller of the hybrid vehicle according to the principles of thepresent disclosure;

FIG. 3 is an example equivalent circuit model for a battery system ofthe hybrid vehicle according to the principles of the presentdisclosure;

FIG. 4A is a functional block diagram of a first example architecture ofa mixed sigma-point Kalman filtering (SPKF) and recursive least squares(RLS) technique according to the principles of the present disclosure;and

FIG. 4B is a flow diagram of a first example method of estimatingbattery system states/parameters according to the principles of thepresent disclosure;

FIG. 5A is a functional block diagram of a second example architectureof the mixed SPKF-RLS technique according to the principles of thepresent disclosure; and

FIG. 5B is a flow diagram of a second example method of estimatingbattery system states/parameters according to the principles of thepresent disclosure.

DETAILED DESCRIPTION

As previously mentioned, there remains a need for an accurate and easilytunable battery management system (BMS). Battery state and parameterestimation performance could be improved by updating battery modelparameters online. Conventional Kalman filter-type methods (e.g.,extended Kalman filtering, or EKF) utilize linear battery models. Tofurther improve estimation performance, however, the non-linearities inthe battery model could be handled by the estimation algorithms.Accordingly, an improved BMS and an improved battery state and parameterestimation technique utilizing mixed sigma-point Kalman filtering (SPKF)and recursive least squares (RLS) are presented. SPKF estimates statesof the battery system (state-of-charge (SOC), voltage, etc.) and ispaired with recursive least squares (RLS) to achieve parameteridentification of the equivalent circuit model parameters.

SPKF provides much better performance than other similar techniques(e.g., EKF using a first order Taylor extension to linearize non-linearfunctions to deal with the non-linearity), particularly when dealingwith a non-linear model. Due to the difficulty of modeling process noisetuning of Kalman filter-type methods, including SPKF, RLS represents agood alternative to estimate such parameters since it uses alinear-in-parameter measurement only model filter structure and a singletunable filter variable. Other benefits include charge/dischargeasymmetry, non-linear hysteresis, self-discharge, non-linear resistanceterms, multi-scaling (see below), and state/parameter estimatedmixing/blending (e.g., when RLS and SPKF both estimate the same term).In other words, the advantages of SPKF and RLS are used to compensatefor each other's disadvantages.

Example advantages include the robustness and accuracy of SOC estimationcapability from SPKF and efficient on-line batter internal parameterestimation capability from RLS that are then used for SOP estimation. Insome implementations, the mixed SPKF-RLS technique could be multi-scaledby assuming that battery system parameters remain constant during shorttime intervals. More particularly, parameter estimations using RLS couldbe performed on a macro (slow) time scale, whereas state estimationsusing SPKF could be performed on a micro (fast) time scale. The benefitsof the mixed SKPF-RLS technique include, but are not limited to,reducing overall size of the battery system and reducing system cost(e.g., easier tuning/calibration), as well as increased performance andoperational life of the battery system due to increased robustness. Forexample, these techniques could be particularly useful in highly dynamicdriving situations.

Referring now to FIG. 1, an example vehicle 100 is illustrated. In oneexemplary implementation, the vehicle 100 is a hybrid vehicle.Non-limiting examples of the vehicle 100 include an electric vehicle(EV), a hybrid electric vehicle (HEV), a plug-in HEV (PHEV), and anextended range electric vehicle (EREV). It will be appreciated, however,that the techniques of the present disclosure could be extended to anyvehicle including a battery system. The vehicle 100 includes anelectrified powertrain 104 configured to generate torque for adrivetrain 108 to propel the vehicle 100. The electrified powertrain 104includes an electric motor 112 and a battery system 116. The vehicle 100optionally includes an engine 120, which could be used, for example, forrecharging the battery system 116 and/or for generating the torque forthe drivetrain 108. The torque could be generated in response to atorque request received via a driver interface 124.

The battery system 116 is any suitable battery system configured togenerate a current to power the electric motor 112. In one exemplaryimplementation, the battery system 116 includes a plurality oflithium-ion (Li-ion) batteries connected together. The battery system116 has one or more sensors 128 associated therewith, which areconfigured to measure current, voltage, and temperature of the batterysystem 116. A controller 132 is configured to receive these measurementsand control the electric motor 112 based on estimated states/parametersof the battery system 116. The controller 132 is also configured tooutput information to the driver interface (SOC percentage, vehiclerange, etc.). Examples of the controller 132 include anapplication-specific integrated circuit (ASIC) and one or moreprocessors operating in a parallel or distributed architecture plus anon-transitory computer-readable medium (memory) having a set ofinstructions stored thereon for execution by the processor(s). While asingle controller 132 is shown, it will be appreciated that the vehicle100 could include multiple controllers, such as a battery managementsystem controller that performs the functionality herein andcommunicates with another controller, such as an engine control unit(ECU).

Referring now to FIG. 2, a functional block diagram of an exampleconfiguration of the controller 132 is illustrated. The controller 132receives information as input from the one or more sensors 128. In oneexemplary implementation, this information includes voltage V, currentI, and temperature T of the battery system 116. It will be appreciatedthat more or less information could be received from the sensor(s) 128,such as only voltage and current. Both an SPKF module 200 and an RLSmodule 204 are configured to receive and utilize at least some of theseinputs. The SPKF module 200 is configured to perform an SPKF techniqueas discussed herein. The RLS module 204 is configured to perform an RLStechnique as discussed herein. In one exemplary implementation, thesemodules 200, 204 are further configured to utilize the outputs of theother module as additional inputs.

Example outputs of the SPKF module 200 include SOC and voltages(referred to generally as V). Example outputs of the RLS module 204include resistances, capacitances, and time constants (referred togenerally as R, C, and τ, respectively). The outputs of these modules200, 204 are utilized by an SOH/SOP estimator 208 to estimate at leastone of SOH and SOP for the battery system 116. The estimated statesoutput by the SOH/SOP estimator 208, and potentially other states outputby the SPKF module 200, are then utilized by a motor control module 212,which is configured to control the electric motor 116. For example only,the SOP could be utilized by the motor control module 212 to maximizeusage of the electric motor 116 or, in other words, to use the electricmotor 116 as much as possible to generate the desired vehicle torqueinstead of engine 120. While not shown, it will be appreciated that atleast some of these states could be output to other components, such asa malfunction indicator lamp (e.g., when the SOH is lower than acritical threshold) or to the driver interface 124 (e.g., to indicate avehicle range corresponding to the SOC).

Referring now to FIG. 3 and with continued reference to FIG. 2, anequivalent circuit model 300 for a particular cell of the battery system116 is illustrated. While described with respect to a particular cell,it will be appreciated that the model could be extended to model theentire battery system 116 of the techniques discussed herein could berepeated for each cell of the battery system 116. V and I representvoltage and current for the particular cell. V_(O) represents an opencircuit voltage, which is also expressed as a sum of a component V_(SOC)dependent on the SOC (hereinafter z, where V_(SOC)=f(z)) and a voltageV_(Cvo) associated with a self-discharge capacitance C_(Vo), which is anon-linear function of capacitor voltage. R_(SD) represents a largeself-discharge resistance and v_(h) represents a hysteresis voltage,e.g., a unitless state value between −1 and 1 to represent dynamics ofone-state hysteresis. Hysteresis voltage can be modelled as either zerostate or one state hysteresis. Zero state hysteresis is modeled as:V _(h) =V _(h) ^(max) *sgn(I)  (1),where V_(h) ^(max) is a maximum hysteresis magnitude, and I is current.One state hysteresis, on the other hand, is modelled as:

$\begin{matrix}{V_{h} = {V_{h}^{\max} \star v_{h}}} & (2) \\{and} & \; \\{{\frac{{dv}_{h}}{dt} = {{- \gamma_{h}}{I \cdot ( {{{{sgn}(I)} \star v_{h}} + 1} )}}},} & (3)\end{matrix}$where v_(h) is a dynamically varying hysteresis voltage state and γ_(h)is a one-state hysteresis gain factor.

The model 200 further defines n RC pairs having respective dynamicresistances R₁ . . . R_(n), capacitances C₁ . . . C_(n), and voltages V₁. . . V_(n). R₀ ^(d) represents an ohmic discharge resistance and R₀^(c) represents an ohmic charge resistance. These ohmic/dynamiccharge/discharge resistances and capacitances are variable and aregenerally dependent on SOC, temperature, current direction and currentmagnitude. For example, the ohmic/dynamic charge/discharge resistancescould have non-linear forms such as:R _(i) ^(c) =R _(i) ^(0,c) +R _(i) ^(I,c)*log(1+|I|)  (4)andR _(i) ^(c) =R _(i) ^(0,d) +R _(i) ^(I,c)*log(1+|I|)  (5),where i is a subscript value from 0 to n, R_(i) ^(I,c), R_(i) ^(I,d) arecharge/discharge resistance values at zero current, and R_(i) ^(I,c),R_(i) ^(I,d) are charge/discharge non-linear resistance values. Theusage of non-linear resistance models the battery response moreaccurately, particularly at lower temperatures (e.g., below zero degreesCelsius). These resistance values are commonly used to calculate anddefine resistance based on state of health (SOH), which could also bereferred to as SOHR. Calculation of state of power (SOP) heavily relieson resistance estimation, since a higher battery resistance results inlower battery power availability.

The term “state” as used herein refers to an overall state of thebattery system (charge, health, power, etc.) as well as a voltages ofthe battery system 116. Examples of the possible estimated states of thebattery system 116 include V_(i) (the voltage for the i^(th) RC pair, ibeing an integer from 0 to n, which represents the model order and couldbe arbitrarily high), V_(Cvo), z, V_(SOC), V_(O), and v_(h). The term“parameter,” on the other hand, refers to internal parameters of thebattery system 116 (resistances, capacitances, charge/discharge timeconstants, etc.), such as with respect to the equivalent circuit model.Examples of the possible estimated parameters of the battery system 116include R₀ ^(d), R₀ ^(c), R_(SD), C_(i) (the capacitance for the i^(th)RC pair), R_(i) (the resistance for the i^(th) RC pair), and C_(Vo), aswell as the maximum magnitude of the hysteresis voltage V_(h) ^(max) andthe one-state hysteresis gain γ_(h). Further examples of the possibleestimated parameters of the battery system 116 include, for the i^(th)RC pair: linear ohmic discharge resistance R₀ ^(0,d), linear ohmiccharge resistance R₀ ^(0,d), non-linear ohmic discharge resistance R₀^(I,d), non-linear ohmic charge resistance R₀ ^(I,d), dynamic RCdischarge resistance R_(i) ^(d), dynamic RC charge resistance R_(i)^(c), linear dynamic RC discharge resistance R_(i) ^(0,d), lineardynamic RC charge resistance R_(i) ^(0,d), non-linear dynamic RCdischarge resistance R_(i) ^(I,d), non-linear dynamic RC chargeresistance R_(i) ^(I,d), charge capacitance C_(i) ^(c), dischargecapacitance C_(i) ^(d), time constant τ_(i), charge time constant τ_(i)^(c), and discharge time constant τ_(i) ^(d).

A non-linear state-space model for the battery system 116 is:x _(k) =f(x _(k−1) ,u _(k−1),θ_(k−1) ,k−1)+v _(k) y _(k) =h(x _(k) ,u_(k),θ_(k) ,k)+w _(k)  (6),where v_(k) and W_(k) each represent an independent, Gaussian noiseprocess and follow Gaussian distributions N(0,Q) and N(0,R),respectively; x_(k), θ_(k), u_(k), and y_(k) represent state, parameter,input, and state estimation output of the model at time k; and f and hrepresent non-linear functions of a state process model and ameasurement model, respectively. The parameter's model, on the otherhand, is:d _(k) =c _(k) ^(T)θ_(k)+ω_(k)  (7),where ω_(k) represents an independent, Gaussian noise process andfollows Gaussian distribution N(0,W), d_(k) represents state, parameter,input, and state estimation output of the model at time k, and c_(k)^(T)∈

^(n) is a regression vector composed of known/measurable quantities. RLSaims to find, recursively in time, an estimate of θ_(k) that minimizesthe following sum of squared errors:E(k)=Σ_(i=1) ^(k)μ^(k−1) [d _(i) −c _(i) ^(T)θ_(k)]²  (8),where 0<λ<1 is a forgetting factor that diminishes the influence of olddata.

In one exemplary implementation, the SPKF module 200 estimates SOC andRC pair voltage. If state of the model is x_(k)=[z_(k), V_(1,k)]^(T),then the state-space model is:

$\begin{matrix}{x_{k} = {{f( {x_{k\text{-}1},u_{k\text{-}1},\theta_{k\text{-}1},{k - 1}} )} + v_{k\text{-}1}}} & (9) \\{{\begin{bmatrix}z_{k} \\V_{1,k}\end{bmatrix} = {\begin{bmatrix}{z_{k - 1} - \frac{T_{s} \cdot I_{k\text{-}1}}{Capa}} \\{{\phi_{1,{k\text{-}1}}V_{1,{k - 1}}} + {{R_{1,{k\text{-}1}}( {1 - \phi_{1k\text{-}1}} )}I_{k\text{-}1}}}\end{bmatrix} + v_{k}}},} & \; \\{y_{k} = {{V_{o,k}( z_{k} )} - {R_{o,k}I_{k}} - V_{1,k} + w_{k}}} & \;\end{matrix}$where z_(k) represents SOC at time k and

${\phi_{1,k} = {\exp( {- \frac{Ts}{\tau_{1,k}}} )}},{\tau_{1,k} = {R_{1,k}{C_{1,k}.}}}$In one exemplary implementation, the RLS module 204 estimates internalresistance R_(o,k), resistance in RC pair R_(1,k) and time constant inRC pair τ_(k). Battery SOH and SOHR can be calculated by module 208using an equation based on both R_(o,k) and R_(1,k). The elements to befed into the RLS module 204 are selected as follows:θ_(k)=[ϕ_(1,k−1) ,R _(1,k−1)(1−ϕ_(1,k−1))−R _(o,k−1)·ϕ_(1,k−1) ,R_(o,k)]^(T)  (10) andc _(k) =[V _(o,k−1) −V _(k−1) ,I _(k−1) ,I _(k)]^(T)  (11).The measurement term of the RLS technique is:

$\begin{matrix}{\begin{matrix}{d_{k} = {V_{o,k} - V_{k} + {\overset{\_}{\omega}}_{k}}} \\{= {V_{1,k} + {I_{k}R_{o,k}} + {\overset{\_}{\omega}}_{k}}} \\{= {{\phi_{1,{k - 1}}V_{1,{k\text{-}1}}} + {{R_{a,{k\text{-}1}}( {1 - \phi_{1,{k\text{-}1}}} )}I_{k\text{-}1}} + {I_{k}R_{o,k}} + {\overset{\_}{\omega}}_{k}}} \\{= {{\phi_{1,{k - 1}}( {{V_{o,{k\text{-}1}}V_{k - 1}} - {I_{k - 1}R_{o,{k - 1}}}} )} + {{R_{a,{k - 1}}( {1 - \phi_{1,{k - 1}}} )}I_{k - 1}} + {I_{k}R_{o,k}} + {\overset{\_}{\omega}}_{k}}} \\{= {{\phi_{1,{k - 1}}( {V_{o,{k - 1}} - V_{k - 1}} )} + {( {{R_{a,{k - 1}}( {1 - \phi_{1,{k - 1}}} )} - {R_{o,{k - 1}}\phi_{1,{k - 1}}}} \rbrack I_{k - 1}} + {I_{k}R_{o,k}} + {\overset{\_}{\omega}}_{k}}} \\{= {{{c_{k}}^{T}\theta_{k}} + {\overset{\_}{\omega}}_{k}}}\end{matrix}.} & (12)\end{matrix}$The open circuit voltage of the battery model V_(o,k), will be updatedusing look-up table and SOC z_(k). Considering that high calculationfrequency is utilized and parameters don't change very fast, it isassumed that all the parameters at the time (k−1) can be taken as theparameters at time k. Then they can be calculated by the result of RLStechnique as follows:

$\begin{matrix}\{ {\begin{matrix}{\tau_{1,k} = {- \frac{T_{s}}{\ln\lbrack {\theta_{k}(1)} \rbrack}}} \\{R_{1,k} = \frac{{\theta_{k}(2)} + {{\theta_{k}(1)}{\theta_{k}(3)}}}{1 - {\theta_{k}(1)}}} \\{R_{o,k} = {\theta_{k}(3)}}\end{matrix}.}  & (13)\end{matrix}$In one exemplary implementation, the forgetting factor λ is 0.99.

Referring now to FIG. 4A, a functional block diagram of a first examplearchitecture 400 for the mixed SPKF-RLS technique is illustrated. Thisarchitecture 400 represents a single time scale version of the SPKF-RLStechnique and thus may also be referred to as a mixed or dual SPKF-RLStechnique. The architecture generally includes SPKF and RLS time updateblocks 404 and 408, respectively, SPKF and RLS measurement update blocks412 and 416, respectively, and respective unit delays 420, 424. Theequation details for the architecture 400 are listed below. In oneexemplary implementation, block 404 executes equations (15)-(16), block408 executes equations (17)-(18), block 412 executes equations(19)-(20), (23), and (25), and block 416 executes equations (21)-(22),(24), and (26).

$\begin{matrix}\begin{matrix}{{{{Initialization}\text{:}\mspace{14mu}{for}\mspace{14mu} k} = 0},{set}} & \begin{matrix}\begin{matrix}{{{\hat{\theta}}_{0}^{+} = {E\lbrack \theta_{0} \rbrack}},{{\hat{x}}_{0}^{+} = {E\lbrack x_{0} \rbrack}}} \\{\sum_{\overset{\sim}{x},0}^{+}{= {E\lbrack {( {x_{0} - {\hat{x}}_{0}^{+}} )( {x_{0} - {\hat{x}}_{0}^{+}} )^{T}} \rbrack}}}\end{matrix} \\{\sum_{\overset{\sim}{\theta},0}^{+}{= {E\lbrack {( {\theta_{0} - {\hat{\theta}}_{0}^{+}} )( {\theta_{0} - {\hat{\theta}}_{0}^{+}} )^{T}} \rbrack}}}\end{matrix}\end{matrix} & (14) \\{{State}\mspace{14mu}{prediction}\{ \begin{matrix}{\mathcal{X}_{k\text{-}1}^{+} = \{ {x_{k\text{-}1}^{+},{x_{k\text{-}1}^{+} + {\gamma( \sqrt{\sum_{x,{k\text{-}1}}^{+}} )}_{1{st}}},\ldots\mspace{14mu},{{\hat{x}}_{k\text{-}1}^{+} + {\gamma( \sqrt{\sum_{x,{k\text{-}1}}^{+}} )}_{pth}},{x_{k\text{-}1}^{+} -}} } \\ \mspace{146mu}{{\gamma( \sqrt{\sum_{x,{k\text{-}1}}^{+}} )}_{1{st}},\ldots\mspace{14mu},{x_{k\text{-}1}^{+} - {\gamma( \sqrt{\sum_{x,{k\text{-}1}}^{+}} )}_{pth}}} \} \\{\mathcal{X}_{k,i}^{+} = {f( {\mathcal{X}_{{k\text{-}1},i}^{+},{\hat{\theta}}_{k\text{-}1}^{+},{{u_{k\text{-}1}k} - 1}} )}} \\{{\hat{x}}_{k}^{-} = {\sum_{i = 0}^{2\mspace{11mu} p}{a_{i}^{(m)}\mathcal{X}_{k,j}^{x, -}}}}\end{matrix} } & (15) \\{{{State}\mspace{14mu}{prediction}\mspace{14mu}{covariance}\mspace{14mu}\sum_{\overset{\sim}{x},k}^{-}} = {{\sum_{i = 0}^{2p}{{\alpha_{i}^{(c)}( {\mathcal{X}_{k,i}^{x, -} - {\hat{x}}_{k}^{-}} )}( {\mathcal{X}_{k,i}^{x, -} - {\hat{x}}_{k}^{-}} )^{T}}} + Q}} & (16) \\{{{Parameter}\mspace{14mu}{prediction}\mspace{14mu}{\hat{\theta}}_{k}^{-}} = {\hat{\theta}}_{k\text{-}1}^{+}} & (17) \\{{{Parameter}\mspace{14mu}{prediction}\mspace{14mu}{covariance}\sum_{\overset{\sim}{\theta},k}^{-}} = \sum_{\overset{\sim}{\theta},{k - 1}}^{+}} & (18) \\{{State}\mspace{14mu}{measurement}\mspace{14mu}{prediction}\{ \begin{matrix}{Y_{k,i} = {h( {\mathcal{X}_{k,i}^{\mathcal{X}, -},u_{k},{\hat{\theta}}_{k}^{-},k} )}} \\{{\hat{y}}_{k} = {\sum\limits_{i = 0}^{2p}{\alpha_{i}^{(m)}Y_{k,i}}}}\end{matrix} } & (19) \\{{State}\mspace{14mu}{filter}\mspace{14mu}{gain}\{ \begin{matrix}{\sum_{\overset{.}{y},k}{= {{\sum_{i = 0}^{2p}{{\alpha_{i}^{(c)}( {Y_{k,i} - {\hat{y}}_{k}} )}( {Y_{k,i} - {\hat{y}}_{k}} )^{T}}} + R}}} \\{\sum_{\overset{\sim}{xy},k}{= {\sum_{i = 0}^{2p}{{\alpha_{i}^{(c)}( {\mathcal{X}_{k,i}^{x, -} - {\hat{x}}_{k}^{-}} )}( {Y_{k,i} - {\hat{y}}_{k}} )^{T}}}}} \\{L_{k}^{x} = {\sum_{\overset{\sim}{xy},k}\sum_{\overset{\sim}{y},k}^{- 1}}}\end{matrix} } & (20) \\{{{Parameter}\mspace{14mu}{measurement}\mspace{14mu}{prediction}\mspace{14mu}{\hat{d}}_{k}} = {c_{k}^{T}{\hat{\theta}}_{k}^{-}}} & (21) \\{{{Parameter}\mspace{14mu}{filter}\mspace{14mu}{gain}\mspace{14mu} L_{k}^{\theta}} = {\sum_{\overset{\sim}{\theta},k}^{-}{c_{k}( {\lambda + {c_{k}^{T}{\sum_{\overset{\sim}{\theta},k}^{-}c_{k}}}} )}^{- 1}}} & (22) \\{{{State}\mspace{14mu}{measurement}\mspace{14mu}{update}\mspace{14mu}{\hat{x}}_{k}^{+}} = {{\hat{x}}_{k}^{-} + {L_{k}^{x}( {y_{k} - {\hat{y}}_{k}} )}}} & (23) \\{{{Parameter}\mspace{14mu}{measurement}\mspace{14mu}{update}\mspace{14mu}{\hat{\theta}}_{k}^{+}} = {{\hat{\theta}}_{k}^{-} + {L_{k}^{\theta}( {d_{k} - {\hat{d}}_{k}} )}}} & (24) \\{{{State}\mspace{14mu}{measurement}\mspace{14mu}{covariance}\mspace{14mu}{update}\sum_{\overset{\sim}{x},k}^{+}} = {\sum_{\overset{\sim}{x},k}^{-}{{- L_{k}^{x}}{\sum_{\overset{\sim}{y},k}( L_{k}^{x} )^{T}}}}} & (25) \\{{{Parameter}\mspace{14mu}{measurement}\mspace{14mu}{covariance}\mspace{14mu}{update}\sum_{\overset{\sim}{\theta},k}^{+}} = {\frac{1}{\lambda}( {1 - {L_{k}^{\theta}c_{k}^{T}}} )\sum_{\hat{\theta},k}^{-}}} & (26)\end{matrix}$

There are various methods to select sigma-point weighting constants. Inone exemplary implementation, central difference Kalman filtering (CDKF)is utilized, which is summarized below:

$\begin{matrix}\; & \gamma & \alpha_{0}^{(m)} & \alpha_{k}^{(m)} & \alpha_{0}^{(c)} & \alpha_{k}^{(c)} \\{C\; D\; K\; F} & h & \frac{h^{2} - L}{h^{2}} & \frac{1}{2h_{2}} & \frac{h^{2} - L}{h^{2}} & \frac{1}{2h^{2}}\end{matrix}$Where h may take any positive value and L represents the state number.For Gaussian, h could equal the square root of 3.

Referring now to FIG. 4B, a flow diagram of a first method 440 ofestimating battery system states/parameters is illustrated. This firstmethod 440, for example, could correspond to the architecture 400illustrated in FIG. 4A and discussed above. Variables to estimate aredefined at 442 and divided for SPKF/RLS at 444. In a first branch,sigma-point calculation occurs at 446 and states are predicted at 448.State prediction co-variance is determined at 450 and state measurementprediction is performed at 454 based on input received at 452. Statefilter gain is determined at 458 based on information received at 456.State measurement covariance and measurement are updated at 460 and 462,respectively, after which a unit delay occurs at 464 and the processreturns to 446. In a second branch, parameter prediction occurs at 466and parameter measurement prediction occurs at 468. Parameter filtergain is determined at 470 based on the information received at 456 andused to update parameter measurement and measurement covariance at 472and 474, respectively. A unit delay occurs at 476 and then the parameterprediction covariance is determined at 476, which is fed back into therelated update block 474.

Referring now to FIG. 5A, a functional block diagram of a second examplearchitecture 500 for the mixed SPKF-RLS technique is illustrated. Thisarchitecture 500 represents a multi-scale version of the SPKF-RLStechnique and thus may also be referred to as a multi-scale mixed ordual SPKF-RLS technique. The architecture generally includes RLS andSPKF time update blocks 504 and 508, respectively, SPKF and RLS timeupdate blocks 512 and 532, respectively, micro and macro unit delays 528and 536, respectively, as well as decision block 516 and time scaletransform blocks 520, 524. The equation details for the architecture 500are listed below. In one exemplary implementation, block 504 executesequations (28)-(29), block 508 executes equations (30)-(31), block 512executes equations (32)-(35), block 520 executes equation (36), andblock 532 executes equations (37)-(40).

$\begin{matrix}{{{{\begin{matrix}{{{{Initialization}\text{:}\mspace{14mu}{for}\mspace{14mu} k} = 0},{set}} & \;\end{matrix}{\hat{\theta}}_{0}^{+}} = {E\lbrack \theta_{0} \rbrack}}\mspace{329mu}{{\hat{x}}_{0,0}^{+} = {E\lbrack x_{0,0} \rbrack}}}\mspace{329mu}{\sum_{\overset{\_}{x},0,0}^{+}{= {E\lbrack {( {x_{0,0} - {\hat{x}}_{0,0}^{+}} )( {x_{0,0} - {\hat{x}}_{0,0}^{+}} )^{T}} \rbrack}}}\mspace{329mu}{\sum_{\overset{\sim}{\theta},0}^{+}{= {E\lbrack {( {\theta_{0} - {\hat{\theta}}_{0}^{+}} )( {\theta_{0} - {\hat{\theta}}_{0}^{+}} )^{T}} \rbrack}}}} & (27) \\{{{Parameter}\mspace{14mu}{prediction}\mspace{14mu}{with}\mspace{14mu}{macro}\mspace{14mu}{scale}\mspace{14mu}{\hat{\theta}}_{k}^{-}} = {\hat{\theta}}_{k\text{-}1}^{+}} & (28) \\{{{Parameter}\mspace{14mu}{prediction}\mspace{14mu}{covariance}\mspace{14mu}{with}\mspace{14mu}{macro}\mspace{14mu}{scale}\mspace{14mu}\sum_{\overset{\sim}{\theta},k}^{-}} = \sum_{\overset{\sim}{\theta},{k - 1}}^{+}} & (29) \\{{State}\mspace{14mu}{prediction}\mspace{14mu}{with}\mspace{14mu}{micro}\mspace{14mu}{scale}\mspace{14mu}\{ \begin{matrix}{\mathcal{X}_{{k\text{-}1},{l - 1}}^{+} = \{ {x_{{k\text{-}1},{l\text{-}1}}^{+},{x_{{k\text{-}1},{l - 1}}^{+} + {\gamma( \sqrt{\sum_{\overset{\_}{x},{k\text{-}1},{l - 1}}^{+}} )}_{1{st}}},\ldots\mspace{14mu},{{\hat{x}}_{{k\text{-}1},{l - 1}}^{+} + {\gamma( {{\sqrt{\sum_{\overset{\_}{x},k}^{+}}{\hat{x}}_{{k\text{-}1},{l\text{-}1}}^{+}} -} }}} } \\ \mspace{146mu}{{\gamma( \sqrt{\sum_{\overset{\sim}{x},{k\text{-}1},{l - 1}}^{+}} )}_{1{st}},\ldots\mspace{14mu},{x_{{k\text{-}1},{l - 1}}^{+} - {\gamma( \sqrt{\sum_{\overset{\_}{x},{k\text{-}1},{l - 1}}^{+}} )}_{pth}}} \} \\{\mathcal{X}_{{k\text{-}1},i,l}^{x, -} = {f( {\mathcal{X}_{{k\text{-}1i},{l - 1}}^{+},{\hat{\theta}}_{k\text{-}1}^{+},{{u_{{k\text{-}1},{l - 1}}k} - 1},{l - 1}} )}} \\{{\hat{x}}_{{k\text{-}1},l}^{-} = {\sum_{i = 0}^{2\mspace{11mu} p}{a_{i}^{(m)}\mathcal{X}_{{k\text{-}1},i,l}^{+}}}}\end{matrix} } & (30) \\{{{State}\mspace{14mu}{prediction}\mspace{14mu}{covariance}\mspace{14mu}{with}\mspace{14mu}{micro}\mspace{14mu}{scale}\mspace{14mu}\sum_{\overset{\sim}{x},{k\text{-}1},l}^{-}} = {{\sum_{i = 0}^{2p}{{\alpha_{i}^{(c)}( {\mathcal{X}_{{k\text{-}1},i,l}^{x, -} - {\hat{x}}_{{k\text{-}1},l}} )}( {\mathcal{X}_{{k\text{-}1},i,l}^{x, -} - {\hat{x}}_{{k\text{-}1},l}^{-}} )^{T}}} + Q}} & (31) \\{{State}\mspace{14mu}{measurement}\mspace{14mu}{prediction}\mspace{14mu}{with}\mspace{14mu}{micro}\mspace{11mu}{scale}\mspace{14mu}\{ \begin{matrix}{Y_{{k\text{-}1},i,l} = {h( {\mathcal{X}_{{k\text{-}1},i,l}^{\mathcal{X}, -},u_{{k\text{-}1},l},{\hat{\theta}}_{k}^{-},{k - 1},l} )}} \\{{\hat{y}}_{{k\text{-}1},l} = {\sum\limits_{i = 0}^{2p}{\alpha_{i}^{(m)}Y_{{k\text{-}1},i,l}}}}\end{matrix} } & (32) \\{{State}\mspace{14mu}{filter}\mspace{14mu}{gain}\mspace{14mu}{with}\mspace{14mu}{micro}\mspace{14mu}{scale}\mspace{14mu}\{ \begin{matrix}{\sum_{\overset{.}{y},{k\text{-}1},l}{= {{\sum\limits_{i = 0}^{2p}{{\alpha_{i}^{(c)}( {Y_{{k\text{-}1},i,l} - {\hat{y}}_{{k\text{-}1},l}} )}( {Y_{{k\text{-}1},i,l} - {\hat{y}}_{{k\text{-}1},l}} )^{T}}} + R}}} \\{\sum_{\overset{\sim}{xy},{k\text{-}1},l}{= {\sum\limits_{i = 0}^{2p}{{\alpha_{i}^{(c)}( {\mathcal{X}_{{k\text{-}1},i,l}^{x, -} - {\hat{x}}_{{k\text{-}1},l}^{-}} )}( {Y_{{k\text{-}1},i,l} - {\hat{y}}_{{k\text{-}1},l}} )^{T}}}}} \\{L_{{k\text{-}1},l}^{x} = {\sum_{\overset{\sim}{xy},{k\text{-}1},l}\sum_{\overset{\sim}{y},{k\text{-}1},l}^{- 1}}}\end{matrix} } & (33) \\{{{State}\mspace{14mu}{measurement}\mspace{14mu}{update}\mspace{14mu}{with}\mspace{14mu}{micro}\mspace{14mu}{scale}\mspace{14mu}{\hat{x}}_{{k\text{-}1},l}^{+}} = {{\hat{x}}_{{k\text{-}1},l}^{-} + {L_{{k\text{-}1},l}^{x}( {y_{{k\text{-}1},l} - {\hat{y}}_{{k\text{-}1},l}} )}}} & (34) \\{{{State}\mspace{14mu}{measurement}\mspace{14mu}{covariance}\mspace{14mu}{update}\mspace{14mu}{with}\mspace{14mu}{micro}\mspace{14mu}{scale}\mspace{14mu}\sum_{\overset{\sim}{x},{k\text{-}1},l}^{+}} = {\sum_{\overset{\sim}{x},{k\text{-}1},l}^{-}{{- L_{{k\text{-}1},l}^{x}}{\sum_{\overset{\sim}{y},{k\text{-}1},l}( L_{{k\text{-}1},l}^{x} )^{T}}}}} & (35) \\{{{{Time}\mspace{14mu}{scale}\mspace{14mu}{transform}\mspace{14mu}{when}\mspace{14mu} l} = {{L\mspace{14mu}{\hat{x}}_{k,0}^{+}} = {\hat{x}}_{{k - 1},L}^{+}}},{\sum_{\overset{\sim}{x},k,0}^{+}{= \sum_{\overset{\sim}{x},{k\text{-}1},L}^{+}}}} & (36) \\{{{Parameter}\mspace{14mu}{measurement}\mspace{14mu}{prediction}\mspace{14mu}{with}\mspace{14mu}{macro}\mspace{14mu}{scale}\mspace{20mu}{\hat{d}}_{k}} = {c_{k}^{T}{\hat{\theta}}_{k}^{-}}} & (37) \\{{{Parameter}\mspace{14mu}{filter}\mspace{14mu}{gain}\mspace{14mu}{with}\mspace{14mu}{macro}\mspace{14mu}{scale}\mspace{14mu} L_{k}^{\theta}} = {\sum_{\overset{\sim}{\theta},k}^{-}{c_{k}( {\lambda + {c_{k}^{T}{\sum_{\overset{\sim}{\theta},k}^{-}c_{k}}}} )}^{- 1}}} & (38) \\{{{Parameter}\mspace{14mu}{measurement}\mspace{14mu}{update}\mspace{14mu}{with}\mspace{14mu}{macro}\mspace{14mu}{scale}\mspace{20mu}{\hat{\theta}}_{k}^{+}} = {{\hat{\theta}}_{k}^{-} + {L_{k}^{\theta}( {d_{k} - {\hat{d}}_{k}} )}}} & (39) \\{{{Parameter}\mspace{14mu}{measurement}\mspace{14mu}{covariance}\mspace{14mu}{update}\mspace{14mu}{with}\mspace{14mu}{macro}\mspace{14mu}{scale}\mspace{14mu}\sum_{\overset{\sim}{\theta},k}^{+}} = {\frac{1}{\lambda}( {1 - {L_{k}^{\theta}c_{k}^{T}}} )\sum_{\hat{\theta},k}^{-}}} & (40)\end{matrix}$

Referring now to FIG. 5B, a flow diagram of a second method 540 ofestimating battery system states/parameters is illustrated. This secondmethod 540, for example, could correspond to the second architecture 500illustrated in FIG. 5A and discussed above. Since the capacity changesvery slowly, it is sufficient to be recalibrated once at several days orweeks, and thus it can be updated at each macro time-step. Variables toestimate are defined at 542 and divided for SPKF and RLS at 544. In afirst branch, SPKF is performed at an SPKF rate at 546. Inputs to 546included tuned filter matrices (548), a micro time-step (550),state-space batter model matrices and vectors (554), and a micro unitdelay (558). Voltage, current, and temperature measurements are obtainedat 552 and fed into 554 along with fixed battery model parametersobtained at 556. The micro unit delay at 558 is obtained by up-samplingan output of an RLS block (562) at 560. In a second branch, RLS isperformed at an RLS rate at 562. The SPKF rate will generally be fasterthan the RLS rate, but it will be appreciated that RLS could beperformed more often than SPKF. Inputs to the RLS block 562 include amacro time-scale (568), a tuned forgetting factor (566), and a macrounit delay (568). The macro unit delay at 568 is obtained bydown-sampling the output of the SPKF block (546) at 570. The outputs ofboth the SPKF block 546 and the RLS block 562 are reported at 564 andused, for example, to calculate other values (SOH, SOP, etc.).

The following Tables 1-2 provide further evidence as to the decreasedcomputational complexity and easier tuning of the mixed or dual SPKF-RLStechniques (both single time interval and multi-scale) according to thepresent disclosure. As can be seen, the computational complexity isorders of magnitude smaller for these techniques compared to othertechniques and there are also many fewer tunable variables.

TABLE 1 Example Computational Complexity Comparison # of stateComputational Complexity^(a) and Dual Multi-scale Dual parameters JointDual SPKF-RLS^(b) SPKF-RLS^(c) variables SPKF^(b) SPKF^(b) (proposed)(proposed) 4 640 160 120 84 6 2160 540 360 279 8 5120 1280 800 656 1010000 2500 1500 1275 n f · n³ 2 · f · (n − 1)² f · ((n − 1)² + f₁ · (n −1)² + (n − 1)³) f₂ · (n − 1)³ ^(a)same number of state and parametervariables, each n_(s) = n_(p) = n/2 ^(b)same state/parameter estimationfrequency of f = 10 ^(c)state estimation frequency f₁ = 10, parameterestimation frequency f₂ = 1

TABLE 2 Example Comparison of Number Tunable Time-Varying FilterVariables # of state and Number of Tunable Time-Varying FilterVariables * parameters Joint Dual Dual SPKF-RLS variables SPKF SPKF(proposed) 2 5 3 3 4 17 9 6 6 37 19 11 8 65 33 18 10 101 51 27 n n² + 12(n − 1)² + 1 (n − 1)² + 2 * same number of state and parametervariables, each n_(s) = n_(p) = n/2

It should be understood that the mixing and matching of features,elements, methodologies and/or functions between various examples may beexpressly contemplated herein so that one skilled in the art wouldappreciate from the present teachings that features, elements and/orfunctions of one example may be incorporated into another example asappropriate, unless described otherwise above.

What is claimed is:
 1. A battery management system for an electrifiedpowertrain of a hybrid vehicle, the system comprising: one or moresensors configured to measure voltage, current, and temperature for abattery system of the hybrid vehicle; and a controller configured to:obtain an equivalent circuit model for the battery system; determine aset of states for the battery system to be estimated; determine a set ofparameters for the battery system to be estimated; receive, from the oneor more sensors, the measured voltage, current, and temperature for thebattery system; using (i) the equivalent circuit model, (ii) themeasured voltage, current, and temperature of the battery system, and(iii) a mixed sigma-point Kalman filtering (SPKF) and recursive leastsquares (RLS) technique to estimate the sets of states and parametersfor the battery system; and using the sets of estimated states andparameters for the battery system to control an electric motor of theelectrified powertrain.
 2. The system of claim 1, wherein the controlleris configured to estimate: the set of states for the battery systemusing an SPKF technique; and the set of parameters for the batterysystem using an RLS technique.
 3. The system of claim 2, wherein thecontroller is further configured to estimate: the set of parameters forthe battery system estimated using the RLS technique are an input forthe SPKF technique; and the set of states for the battery systemestimated using the SPKF technique are an input for the RLS technique.4. The system of claim 1, wherein the set of states includes at leastone of a state of charge (SOC) of the battery system and a voltage ofthe battery system.
 5. The system of claim 2, wherein the SPKF techniqueand the RLS technique are both configured to estimate a particular stateor parameter for the battery system, and wherein the controller isconfigured to estimate the particular state or parameter of the batterysystem based on at least one of these estimates.
 6. The system of claim5, wherein the controller is further configured to calculate acovariance indicative of a confidence in each estimate of the particularstate or parameter for the battery system, wherein the controller isconfigured to utilize the covariance in estimating the particular stateor parameter of the battery system.
 7. The system of claim 5, whereinthe controller is configured to estimate the particular state orparameter for the battery system based on an average of the estimatesfrom the SPKF and RLS techniques.
 8. The system of claim 2, wherein thecontroller is configured to estimate the set of parameters for thebattery system using the RLS technique based further on a tunedforgetting factor.
 9. The system of claim 1, wherein the set ofparameters includes at least one of a resistance and a charge/dischargetime constant for the battery system.
 10. The system of claim 1, whereinthe equivalent circuit model is an asymmetric equivalent circuit modelhaving charge/discharge asymmetry.
 11. The system of claim 2, whereinthe controller is further configured to: obtain an SPKF rate at which toperform the SPKF technique; obtain an RLS rate at which to perform theRLS technique; and estimate the sets of states and parameters for thebattery system using the SPKF and RLS techniques according to theirrespective rates.
 12. The system of claim 10, wherein the SPKF rate isgreater than the RLS rate.
 13. The system of claim 1, wherein thecontroller is further configured to estimate at least one of a state ofpower (SOP) and a state of health (SOH) for the battery system based onthe sets of estimated states and parameters.
 14. The system of claim 1,wherein the controller is configured to control the electric motor basedfurther on the at least one of the SOP and the SOH for the batterysystem.
 15. The system of claim 1, wherein: the hybrid vehicle comprisesan engine; and the controller is configured to coordinate control of theengine and the electric motor based on the estimated states/parametersto maximize usage of the electric motor.